Applying Kirchhoff's Voltage Law

Hello! Welcome to the fifth lesson in our series on DC circuit fundamentals.

In the last lesson, we explored Kirchhoff's Current Law (KCL), which governs how current behaves at a node based on the principle of charge conservation. You learned that the total current entering a node must equal the total current leaving it.

Today, we'll learn the other foundational law of circuit analysis: Kirchhoff's Voltage Law (KVL). While KCL deals with currents at a point, KVL deals with voltages around a closed path, or loop. It's based on another fundamental principle: the conservation of energy. Your goal for this lesson is to be able to apply KVL to analyze voltage drops and rises around circuit loops.

Together, KCL and KVL form the bedrock of all DC circuit analysis, allowing us to solve for the voltages and currents in any network of components, no matter how complex.

1. The Principle of KVL: Conservation of Energy

Imagine walking on a hilly terrain. If you start at a certain point, walk around any path, and return to that exact same starting point, your net change in elevation will be zero. You might go up and down, but you end where you began.

Voltage, or electric potential, is analogous to elevation. Kirchhoff's Voltage Law formalizes this idea for electric circuits.

Kirchhoff's Voltage Law (KVL) states that the algebraic sum of all voltages around any closed loop in a circuit must equal zero.

This is a direct consequence of the law of conservation of energy. If a charge moves around a closed loop and returns to its starting point, its net change in potential energy must be zero. Since voltage is potential energy per unit charge ( $V = E / Q$ ), the net change in voltage must also be zero.

Let's watch a short video that introduces this concept.

Kirchhoff's Voltage Law (KVL)

This clip from Neso Academy clearly states KVL and, importantly, explains its basis in the law of conservation of energy.

First, watch from 00:00 to 00:46 to get the formal statement of the law. Then, watch from 08:15 to 09:12 to understand its connection to the conservation of energy.

2. The Rules of the Road: Sign Conventions for KVL

To apply KVL, we need a consistent method for determining the sign (positive or negative) of each voltage term in our sum. A voltage source adds energy to the circuit (a voltage rise), while a resistor dissipates energy as current flows through it (a voltage drop).

Formulations of Kirchhoff's Voltage Law with Circuit Example — This image shows three equivalent ways to state KVL. We will primarily use Formulation 2: "The algebraic sum of voltage drops and rises around a loop is zero."

Here is a systematic procedure we will follow:

Assume a current direction for each loop in the circuit. It's okay if you guess wrong; the math will simply produce a negative result for the current, indicating it flows in the opposite direction.
Assign voltage polarity (+/-) across resistors. Current always flows from higher potential (+) to lower potential (-) through a resistor. So, for your assumed current direction, mark the side the current enters as '+' and the side it exits as '-'. Voltage sources have their polarity fixed.
Pick a starting point and a direction of travel around the loop (e.g., clockwise).
Walk around the loop and write down the voltage terms:
- If you cross a component from the - terminal to the + terminal, it's a voltage rise. Add this voltage to your sum (e.g., +V).
- If you cross a component from the + terminal to the - terminal, it's a voltage drop. Subtract this voltage from your sum (e.g., -V).
Once you return to your starting point, set the entire sum equal to zero.

This process can be tricky at first. Let's watch a detailed video that explains these sign conventions very clearly.

Kirchhoff's Law, Junction & Loop Rule, Ohm's Law - KCl & KVl Circuit Analysis - Physics

The Organic Chemistry Tutor provides a fantastic, methodical explanation of the sign conventions for both batteries and resistors, which is the key to correctly applying KVL.

Watch the segment from 01:50 to 06:40. Pay close attention to how the direction of travel around the loop and the direction of current through a resistor determine whether the voltage term is positive (a rise) or negative (a drop).

3. Applying KVL to a Simple Series Circuit

Let's put this into practice with a basic series circuit. We'll use KVL to find the current flowing in the circuit and the voltage drop across each resistor.

For this example, let's use the procedure outlined above on the circuit provided in this reading.

Kirchoff's Laws and Circuit Analysis

The article 'Kirchoff's Laws and Circuit Analysis' gives a clear, step-by-step example of creating a KVL equation for a simple series circuit. It also shows that your choice of starting point or direction of travel doesn't change the final answer.

Read the sections 'Example' (under the 'KVL' heading) and 'Series Circuits'. Focus on the example where they build the KVL equation V₀ - V₁ - V₂ = 0. Then, follow the solved problem where they use this KVL equation along with Ohm's Law (V=IR) to find all unknown currents and voltages.

Let's recap the process from the reading for the final solved example:

KVL Equation: For the series circuit with source $V_{0}$ and resistors $R_{1}, R_{2}$ , the KVL equation is $V_{0} - V_{1} - V_{2} = 0$ .
Ohm's Law Substitution: We know $V_{1} = I_{0} R_{1}$ and $V_{2} = I_{0} R_{2}$ . Substituting these into the KVL equation gives $V_{0} - I_{0} R_{1} - I_{0} R_{2} = 0$ .
Solve for Current: Rearranging the equation gives $V_{0} = I_{0} (R_{1} + R_{2})$ , which allows us to solve for the unknown current $I_{0}$ .
Find Voltage Drops: Once $I_{0}$ is known, we can find the individual voltage drops $V_{1}$ and $V_{2}$ using Ohm's Law.

This demonstrates how KVL and Ohm's Law work together to fully analyze a circuit.

Test your understanding!

Consider a simple loop with a 12V battery and two resistors, $R_{1} = 10Ω$ and $R_{2} = 2Ω$ , connected in series.

Write the KVL equation for the loop.
Calculate the current $I$ flowing in the loop.
Calculate the voltage drops across $R_{1}$ and $R_{2}$ . Do they sum to the source voltage?

Show answer

KVL Equation: Let's assume current $I$ flows out of the positive terminal of the battery. We start at the negative terminal and move clockwise.
- Rise across the battery: $+ 12 V$
- Drop across $R_{1}$ : $- V_{1} = - I \cdot R_{1}$
- Drop across $R_{2}$ : $- V_{2} = - I \cdot R_{2}$
  The equation is: $12 - I \cdot R_{1} - I \cdot R_{2} = 0$
Calculate Current:
$12 - I (10) - I (2) = 0$
$12 - 12 I = 0$
$12 I = 12 ⟹ I = 1 A$
Calculate Voltage Drops:
- $V_{1} = I \cdot R_{1} = 1 A \cdot 10Ω = 10 V$
- $V_{2} = I \cdot R_{2} = 1 A \cdot 2Ω = 2 V$
  The sum of the drops is $10 V + 2 V = 12 V$ , which equals the source voltage, confirming our analysis.

4. KVL in Multi-Loop Circuits

The real power of KVL becomes apparent in circuits with multiple loops and sources, where simple series/parallel rules don't apply. For these circuits, we apply KVL to each independent loop. This, often combined with KCL at the nodes, creates a system of simultaneous linear equations. Since you're studying linear algebra, you know that such systems can be solved to find all the unknown currents.

This systematic application of KVL is called Mesh Analysis, a topic we will cover in detail in the next module. For now, let's see an example of how the equations are set up and solved.

Kirchhoff's Law, Junction & Loop Rule, Ohm's Law - KCl & KVl Circuit Analysis - Physics

Let's return to The Organic Chemistry Tutor. This next section applies KVL to a two-loop circuit, showing how to create two separate equations and solve them simultaneously.

Watch the segment from 06:40 to 15:20. Notice how he sets up a KCL equation at the junction (i₁ = i₂ + i₃), then writes a KVL equation for Loop 1 and another for Loop 2. He then solves this system of equations to find all the currents.

This example perfectly illustrates the general strategy: for a circuit with N independent loops, you'll typically need N KVL equations (and possibly some KCL equations) to solve for all the unknown currents.

5. A Powerful Application: Finding Voltage Between Any Two Points

KVL isn't just for closed loops. You can use its principles to find the potential difference (voltage) between any two points in a circuit. The method is simple:

Start at your first point (e.g., point A).
Trace any path through the circuit to your second point (e.g., point B).
Algebraically sum the voltage rises and drops you encounter along that path.
The final sum is the potential difference between the two points, $V_{A B}$ .

Amazingly, the path you choose doesn't matter; the result will always be the same.

Kirchhoff's Voltage Law (KVL)

This is a non-obvious but extremely useful technique. The Neso Academy video provides an excellent walkthrough of how to use KVL to find the voltage between two arbitrary points, and demonstrates that the result is the same regardless of the path taken.

Watch the segment from 11:19 to 15:21. Follow along as he calculates the potential difference VA - VB using two different paths and arrives at the same KVL equation both times. This is a very powerful concept.

Conclusion

You have now mastered the two fundamental laws of circuit analysis. KCL governs the flow of charge into and out of nodes, while KVL governs the energy changes in a closed loop.

Key Takeaways:

Kirchhoff's Voltage Law (KVL) is a statement of the conservation of energy.
It states that the algebraic sum of all voltages around a closed loop is zero.
A consistent sign convention is crucial for applying KVL correctly. Voltage rises are typically positive, and voltage drops are negative.
By applying KVL to each loop in a circuit, we can create a system of linear equations to solve for all unknown currents.

In the next lesson, we will see how KVL and KCL lead directly to the formulas for calculating the equivalent resistance of resistors connected in series and in parallel. This will give us our first major shortcut for simplifying and analyzing circuits.